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‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Foundations of Modern MacroeconomicsThird Edition
Chapter 11: New Keynesian economics
Ben J. Heijdra
Department of Economics, Econometrics & FinanceUniversity of Groningen
13 December 2016
Foundations of Modern Macroeconomics - Third Edition Chapter 11 1 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Outline
1 ‘Keynesian’ MultipliersA basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
2 Monopolistic competition and moneyA basic monetary monopolistic competititon modelMonetary neutrality
3 Imperfectly flexible pricesNon-convex adjustment costsConvex adjustment costs
Foundations of Modern Macroeconomics - Third Edition Chapter 11 2 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Aims of this chapter
Monopolistic competition as a micro-foundation for themultiplier (is it Keynesian?)
Monopolistic competition and welfare-theoretic aspects
The marginal costs of public funds and the multiplier
Monetary non-neutrality and price adjustment costs
Nominal and real rigidity: definitions and interaction
Foundations of Modern Macroeconomics - Third Edition Chapter 11 3 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
The “Keynesian multiplier”
Literature is related to the quantity rationing approach of the1970s and 1980s
Key question: Who sets the prices?
Auctioneer? Fictional deus ex machina
Price setting firms? Monopolistic competition
Develop simple static macro model with monopolisticcompetition
Study two cases:
Flexible pricesSticky prices
Foundations of Modern Macroeconomics - Third Edition Chapter 11 4 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
A static model of monopolistic competition
Households, many small firms, government
Horizontal product differentiation
Single production factor: labour
Foundations of Modern Macroeconomics - Third Edition Chapter 11 6 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Representative household
Household utility:
U ≡ Cα(1− L)1−α, 0 < α < 1
C is composite consumptionL is labour supply (1− L is leisure)U is utility
Foundations of Modern Macroeconomics - Third Edition Chapter 11 7 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Representative household
Composite consumption: S-D-S preferences:
C ≡ Nη
N−1N∑
j=1
Cj(θ−1)/θ
θ/(θ−1)
N is number of product varietiesCj is variety j∞ ≫ θ > 1: close but imperfect substitutesη ≥ 1: preference for diversity (“taste for variety”) Spreadgiven production over as many varieties as possible if η > 1
Foundations of Modern Macroeconomics - Third Edition Chapter 11 8 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Representative household
Household budget constraint:
N∑
j=1
PjCj = WL+Π− T
Pj is price of variety jW is nominal wage rateΠ is profit income of the household (from MC sector)T is lump-sum taxes
Household chooses L, Cj (for j = 1, 2, · · · , N) to maximizeU subject to the budget constraint and taking as given Wand Pj (for j = 1, 2, . . . , N)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 9 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Representative household
Solutions:
PC = α [W +Π− T ]
W (1− L) = (1− α) [W +Π− T ]
Cj
C= N−(θ+η)+ηθ
(Pj
P
)−θ
(j = 1, · · · , N)
P is a true price index depending on the Pj ’s and on N :
P ≡ N−η
N−θN∑
j=1
Pj1−θ
1/(1−θ)
W +Π− T is full income
CD preferences imply constant spending sharesDemand for variety j is price elastic (θ is the elasticity)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 10 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Representative firm
Technology:
Yj =
0 if Lj ≤ FLj−F
k if Lj ≥ F
Yj is output of firm j (producing variety j)Lj is labour used by firm j1/k is the (constant) marginal product of labourF > 0 is fixed costs (“overhead labour”): increasing returns toscale at firm level
Foundations of Modern Macroeconomics - Third Edition Chapter 11 11 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Representative firm
Profit definition:
Πj ≡ PjYj −W [kYj + F ]
Πj is profit of firm jPjYj is revenue of firm jWLj = W [kYj + F ] is costs of firm j
We anticipate that Pj depends on output by firm j (and oncompetitors’ output), Pj = Pj (Yj), and adopt the Cournotassumption (firm j takes other firms’ output as given)
The choice problem is:
MaxYj
Πj = Pj(Yj)Yj −W [kYj + F ]
Foundations of Modern Macroeconomics - Third Edition Chapter 11 12 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Representative firm
The optimal decision rule is:
dΠj
dYj= Pj + Yj ·
∂Pj
∂Yj−Wk = 0 ⇒
Pj = µjWk (a)
(a): price is set equal to a gross markup, µj , times marginal(labour) cost, WkThe gross markup is:
µj ≡εj
εj − 1, εj ≡ −
∂Yj
∂Pj
Pj
Yj
The higher is εj , the lower is µj (lower market power)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 13 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Government
Levies lump-sum tax, T , on household
Employs (useless) civil servants, LG
Consumes a composite good, G, defined analogously to C:
G ≡ Nη
N−1N∑
j=1
Gj(θ−1)/θ
θ/(θ−1)
η same as in C: price index sameθ same as in C: price elasticity same
Foundations of Modern Macroeconomics - Third Edition Chapter 11 14 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Government
Assume government is a cost minimizer: chooses Gj (forj = 1, 2, · · · , N) in order to “produce” a given level of G atleast cost
Derived government demand for variety j is:
Gj
G= N−(θ+η)+ηθ
(Pj
P
)−θ
(j = 1, · · · , N)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 15 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Some loose ends
Demand facing firm j is:
Yj = Cj +Gj
= (C +G)N−(θ+η)+ηθ
︸ ︷︷ ︸
(a)
(Pj
P
)−θ
︸ ︷︷ ︸
(b)
(a): shift factors affecting firm j’s demand(b): relative price factor affecting firm j’s demand
Demand elasticity is constant (and equal to θ):
µj =θ
θ − 1= µ (for all j = 1, · · · , N)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 16 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Some loose ends
Symmetric model: for all j = 1, · · · , N we have:
Pj = µWk = P , Yj = Y , Lj = L
Aggregate quantity index, Y , is:
Y ≡
∑Nj=1 PjYj
P
Labour market equilibrium (LME):
L = LG +
N∑
j=1
Lj
Summary of the model is provided in Table 11.1. (Briefly runthrough table; LME implied by model via Walras Law)
We can use W as the numeraire (everything measured inwage units)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 17 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Table 11.1: A simple macro model with monopolisticcompetition
Y = C +G (T1.1)
PC = αIF , IF ≡ [W +Π− T ] (T1.2)
Π ≡
N∑
j=1
Πj =1
θPY −WNF (T1.3)
T = PG+WLG (T1.4)
P = N1−ηP = N1−ηµWk (T1.5)
W (1− L) = (1− α)IF (T1.6)
V =IFPV
, PV =
(P
α
)α (W
1− α
)1−α
(T1.7)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 18 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Balanced-budget multiplier
Short-run multiplier (N fixed–no entry of new firms)
Financed with lump-sum taxes, T ↑Financed by firing civil servants, LG ↓ (proxy for “bondfinancing” in a static model)
Long-run multiplier (N variable–free entry of firms)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 20 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Short-run multiplier
N = N0 (fixed); GBC: dG = d (T/P )
Aggregate consumption function is:
C = α [1−N0F − LG]W︸ ︷︷ ︸
c0
+α
θY − αG
c0 is fixed in the short runW/P is fixed in the short run (in the SE P depends on N only)C depends on Y via the profit channel (as Y ↑ ⇒ aggregateprofit income, Π ↑ ⇒ C ↑ (and (1− L) ↑)G effect is due to taxation (G ↑ ⇒ T ↑, C ↓ (and (1− L) ↓)
The effect of an increase in G is illustrated in Figure 11.1
Foundations of Modern Macroeconomics - Third Edition Chapter 11 21 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Figure 11.1: Government spending multiplier
C,Y
Y
!!
E0
Y = C + G
G1
G0
C = c0 + (α/θ)Y ! αG1
Y = C + G1
E1
!
Y1Y0
C0
C1
C = c0 + (α/θ)Y ! αG0
Y = C + G0
!
Foundations of Modern Macroeconomics - Third Edition Chapter 11 22 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Short-run multiplier I
Effect on output:(dY
dG
)SR
T
=
(θdΠ
PdG
)SR
T
= (1− α)
[
1 +∞∑
i=1
(α/θ)i
]
=1− α
1− α/θ> 1− α
Degree of monopoly, 1θ , does magnify the expansionary effect!
Effect on consumption:
−α <
(dC
dG
)SR
T
= −θ − 1
θ − αα < 0
Inconsistent with Haavelmo b-b multiplier (where C isunaffected)!
Foundations of Modern Macroeconomics - Third Edition Chapter 11 23 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Short-run multiplier I
Effect on employment:
0 < W
(dL
dG
)SR
T
=θ − 1
θ − α(1− α) < 1− α
Labour supply effect explains output expansion (ratherclassical mechanism)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 24 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Short-run multiplier II
N = N0 (fixed); GBC: dG = −WdLG
Aggregate consumption function is:
C = α [1−N0F ]W +α
θY − α
T
P
T/P is constant (by assumption)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 25 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Short run multiplier II
Effects of increase in government consumption:
(dY
dG
)SR
LG
=
(θdΠ
PdG
)SR
LG
=
[
1 +∞∑
i=1
(α/θ)i
]
=1
1− α/θ> 1
(dC
dG
)SR
LG
=α
θ − α> 0
W
(dL
dG
)SR
LG
= −1− α
θ − α< 0
dYdG exceeds unity as consumption rises
(dCdG > 0
)!
Labour supply falls (wealth effect) but output expansion madepossible by release of labour from the unproductive to theproductive sector
Foundations of Modern Macroeconomics - Third Edition Chapter 11 26 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Long-run multiplier
Following a fiscal shock there are excess profits to be gained(Π > 0)
In absence of barriers to entry one would expect entry of newfirms
Ad hoc entry/exit rule:
N = γN (Π/P ) = γN[θ−1Y −WNF
], γN > 0
What is the long-run multiplier? Assume there are no civilservants (LG = 0)
Goods market equilibrium (GME) line:
Y = α [1−NF ]W + (α/θ)Y + (1− α)G
=
[α(1−NF )
µk(1− α/θ)
]
Nη−1 +
[1− α
1− α/θ
]
G (GME)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 27 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Long-run multiplier
ContinuedWe have used the pricing rule:
W =Nη−1
µk
The zero-profit (ZP) condition is:
Y =θFNη
µk(ZP)
In Figure 11.2 we illustrate the impact, transitional, andlong-run effect of a tax-financed increase in governmentconsumption
ZP slopes upThere is entry (exit) of firms to the left (right) of the ZP line(see the horizontal arrows)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 28 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Long-run multiplier
Slope of GME is ambiguous due to interplay of offsettingeffects
Diversity effect if η > 1: renders slope positiveFixed-cost effect (for F > 0): renders the slope negative
Two special cases for GME:
Standard S-D-S preferences (set η = µ): GME slopes up (seeFigure 11.2). Long-run multiplier is larger than the short-runmultiplier:
(dY
dG
)LR,η=µ
T
=1− α
1− µ−1µ [α+ (1− α)ωC ]
=1− α
1− (1/θ) [α+ (1− α)ωC ]>
1− α
1− α/θ≡
(dY
dG
)SR
T
Foundations of Modern Macroeconomics - Third Edition Chapter 11 29 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Long-run multiplier
ContinuedNo PFD at all (set η = 1): GME slopes down. Long-runmultiplier “vanishes”:
0 <
(dY
dG
)LR,η=1
T
= (1− α) <1− α
1− α/θ≡
(dY
dG
)SR
T
Diversity effect shows up in the “aggregate productionfunction” for this economy, relating Y to L:
Y =(θF )1−η
µkLη
η > 1 implies IRTS at the aggregate levelSome hard-core Keynesians argue that IRTS are (or should be)the central element of Keynesian economics (PFD is onesimple mechanism)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 30 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Figure 11.2: Multipliers and firm entry
!
!
E0
W
!
!
E0=E1
!
E1
Y
N
N
E2
E2
N0 N1
W0
W1
Y0
ZP
GME1
GME0
Foundations of Modern Macroeconomics - Third Edition Chapter 11 31 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Welfare effects
Establish link between the multiplier and welfare
Look at short-run multiplier only
Handy tool – the indirect utility function (IUF):
V ≡IFPV
≡W +Π/P − T/P
PV /P
PV
P≡
W 1−α
αα(1− α)1−α
Foundations of Modern Macroeconomics - Third Edition Chapter 11 33 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Tax-financed fiscal policy
Substitute GBC and profit definition into IUF:
V ≡[1−NF − LG]W + (1/θ)Y −G
PV /P
Recall: N , W , P , and PV all fixed in the short runV rises with Y : output too low from a social point of viewV falls with G: taxation hurts
Differentiate V with respect to G:
(dV
dG
)SR
T
=P
PV
[
1
θ
(dY
dG
)SR
T
− 1
]
= −P
PV
θ − 1
θ − α< 0
Foundations of Modern Macroeconomics - Third Edition Chapter 11 34 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Tax-financed fiscal policy
Fiscal policy is not welfare increasing (contra Keynes claimabout empty bottles). Reasons for this un-Keynesian result:
Flexible wages and clearing labour marketEvery unit of labour is productive
Let us reconsider the bond-financed case again
Foundations of Modern Macroeconomics - Third Edition Chapter 11 35 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Bond-financed fiscal policy
Extra spending financed by firing unproductive civil servants(dG = −WdLG)
For this case the IUF is:
V ≡[1−NF ]W + (1/θ)Y − T/P
PV /P
T/P is fixedOnly output effect remains
Differentiate V with respect to G:
(dV
dG
)SR
LG
=
(P
PV
)1
θ
(dY
dG
)SR
LG
=P
PV
1
θ − α> 0
Foundations of Modern Macroeconomics - Third Edition Chapter 11 36 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic real monopolistic competition modelGovernment spending multipliersWelfare effects of public spending
Bond-financed fiscal policy
Fiscal policy increases welfare!
Labour shifted from unproductive to productive activitiesBut: a tax cut would improve welfare even more:
(dV
d(T/P )
)SR
LG
=P
PV
[
1
θ
(dY
d(T/P )
)SR
LG
− 1
]
=P
PV
θ
θ − α> 0
Foundations of Modern Macroeconomics - Third Edition Chapter 11 37 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic monetary monopolistic competition modelMonetary neutrality
A basic monetary model (1)
Turn from real to monetary model
Usual short-cut trick: put money in the utility function
Money saves on shoe-leather costsShopping costs depend on leisure and moneyMoney makes shopping easier (saves valuable leisure)
Household utility function:
U ≡[Cα(1− L)1−α
]β(M
P
)1−β
, 0 < α, β < 1
where M is nominal money balances
Foundations of Modern Macroeconomics - Third Edition Chapter 11 39 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic monetary monopolistic competition modelMonetary neutrality
A basic monetary model (2)
Household budget constraint:
PC +W (1− L) +M = M0 +W +Π− T
where M0 is initial money balances (accumulated in theprevious period)
Household chooses C, L, and M to maximize U subject tothe budget constraint. Solutions:
PC = αβIF
IF ≡ M0 +W +Π− T
W (1− L) = β(1− α)IF
M = (1− β)IF
Foundations of Modern Macroeconomics - Third Edition Chapter 11 40 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic monetary monopolistic competition modelMonetary neutrality
A basic monetary model (3)
Assume that the policy maker maintains a constant moneysupply. Money market equilibrium (MME) is then:
M = M0
The monetary monopolistic competition model is summarizedin Table 11.2. Some remarks:
M0 features in the indirect utility function (IUF, eqn (T2.8))Helicopter drop of money, dM0 > 0, has no welfare effectsMoney is neutral / classical dichotomydM0 > 0 inflates nominal variables but leaves real variablesunchangedIn and of itself, monopolistic competition does not causemonetary non-neutrality
Foundations of Modern Macroeconomics - Third Edition Chapter 11 41 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic monetary monopolistic competition modelMonetary neutrality
Table 11.2: A simple monetary monopolistic competitionmodel
Y = C +G (T2.1)
C = αβIFP
,IFP
≡
M0
P+
W
P+
Π
P−
T
P(T2.2)
Π
P≡
1
θY −
W
PNF (T2.3)
T
P= G+
W
PLG (T2.4)
P
W= µkN1−η (T2.5)
W
P(1− L) = β(1− α)
IFP
(T2.6)
M0
P= (1− β)
IFP
(T2.7)
V =IFPV
, PV =
(P
αβ
)αβ (W
β(1− α)
)β(1−α) (P
1− β
)1−β
(T2.8)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 42 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic monetary monopolistic competition modelMonetary neutrality
Properties of the monetary monopolistic competitionmodel
Model can be reduced to two schedules
Focus on the short run: N and W are fixed
Goods market equilibrium (GME) locus:
Y =α [1−NF − LG]W + (1− α)G
1− α/θ
Money market equilibrium (MME) locus:
M0
P=
1− β
β
[
[1−NF − LG]W + (1/θ)Y −G]
Classical dichotomy:GME fixes Y independently from M0
MME then fixes P
Foundations of Modern Macroeconomics - Third Edition Chapter 11 44 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
A basic monetary monopolistic competition modelMonetary neutrality
Properties of the monetary monopolistic competitionmodel
Effects on lump-sum tax financed fiscal policy:
0 <
(dY
dG
)SR
T
=1− α
1− α/θ< 1
(dW
W
)SR
T
=
(dP
P
)SR
T
=
(dP
P
)SR
T(dM0/P
dG
)SR
T
= −M0
P 2
(dP
dG
)SR
T
= −(1− β)(θ − 1)
β(θ − α)< 0
Monetary part of the model is more Classical than Keynesian!
Foundations of Modern Macroeconomics - Third Edition Chapter 11 45 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Sticky prices and monetary non-neutrality
Under which conditions would a price-setting agent change hisprice or keep it unchanged?
Key ingredient of the New Keynesian approach: non-trivialprice adjustment costs (remember Modigliani (1944)?)
Two types of price adjustment costs:
Menu costs (non-convex): fixed cost per price change (e.g.informing dealers, reprinting price lists or “menu’s”, etcetera)Convex costs: costs depending on the size of the price change(e.g. adverse reactions by customers to large price changes)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 46 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Menu costs
Develop simplified version of the Blanchard-Kiyotaki model(competitive labour market)
Focus on the short run: fixed number of firms (N)
Household utility is additively separable in (C,M/P ) and L:
U(C,M/P,L) ≡ U1(C,M/P )− U2(L)
= Cα(M/P )1−α − γLL1+1/σ
1 + 1/σ, 0 < α < 1
σ > 0 regulates labour supply elasticityC is composite differentiated good (η = 1: no diversitypreference)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 47 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Menu costs
Household budget restriction:
PC +M = WL+M0 +Π− T (≡ I)
Use two-stage budgeting:
stage 1: maximizing U1(C,M/P ) subject to PC +M = Iyields:
PC = αI
M = (1− α)I
V 1(I/P ) = αα(1− α)1−α(I/P )
Foundations of Modern Macroeconomics - Third Edition Chapter 11 49 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Menu costs
Continuedstage 2: maximizing V 1(I/P ) (the IUF associated with stage1 problem) subject to I ≡ WL+M0 +Π− T yields:
L =
(αα(1− α)1−α
γL
)σ (W
P
)σ
(a)
I
P=
(αα(1− α)1−α
γL
)σ (W
P
)1+σ
+M0 +Π− T
P
Key feature of the labour supply equation (a): no incomeeffect. Substitution effect parameterized by σ. Note:
σ large: near horizontal labour supply equation. Small changein W causes large change in L. High degree of real rigidity(empirically problematic)σ small: near vertical labour supply equation. Small change inL causes large change in W . Low degree of real rigidity(empirically realistic)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 50 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Menu costs
Firms face demand from private sector and from thegovernment (same elasticity; no diversity effect)
Yj(Pj , P, Y ) =
(Pj
P
)−θ Y
N
Aggregate demand is:
Y = C +G =α
1− α·M
P+G
G raises aggregate demandIf P is somehow fixed (e.g. due to menu costs), then M willalso raise aggregate demand
Foundations of Modern Macroeconomics - Third Edition Chapter 11 51 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Menu costs
Technology of the differentiated product firm is slightly moregeneral than before:
Yj =
0 if Lj ≤ F[Lj−F
k
]γif Lj ≥ F
We had γ = 1 but now also allow for 0 < γ < 1γ regulates curvature of the marginal cost curve (γ < 1, MCfalls with output, and AC is U-shaped)
Firm chooses its price, Pj , in order to maximize its profit:
Πj(Pj , P, Y ) ≡ PjYj(Pj , P, Y )︸ ︷︷ ︸
revenue
−W[
k (Yj(Pj , P, Y ))1/γ + F]
︸ ︷︷ ︸
total cost
Bertrand assumption: firm takes prices of close competitors asgiven (P is an aggregate of these prices)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 52 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Menu costs
The optimal price for firm j satisfies the FONC:
dΠj(Pj , P, Y )
dPj= [Pj −MCj ]
∂Yj(Pj , P, Y )
∂Pj+ Yj(Pj , P, Y )
= Yj(Pj , P, Y )
[
1 +Pj −MCj
Pj
Pj
Yj(·)
∂Yj(·)
∂Pj
]
= Yj(Pj , P, Y )
[
1− θPj −MCj
Pj
]
= 0 (a)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 53 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Menu costs
We derive from (a) that the optimal price is a markup timesmarginal cost (MCj), i.e. Pj = µMCj or:
Pj =
(µk
γ
)
WY(1−γ)/γj , µ =
θ
θ − 1> 1
Without menu costs optimal pricing rule under Cournot andBertrand same (not so with menu costs!)Relative price of firm j depends on Yj and on the real wage,W/P :
Pj
P=
µk
γ
W
PY
(1−γ)/γj
This is where the aggregate labour market comes into play
Model is summarized in Table 11.3
Foundations of Modern Macroeconomics - Third Edition Chapter 11 54 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Table 11.3: A simplified Blanchard-Kiyotaki model (nomenu costs)
Y = C +G (T3.1)
C =α
1− α
M0
P=
α
[
ω−σ(
WP
)1+σ+ M0
P+ Π
P−G
]
(if σ < ∞)
α[(
WP
)
L+ M0
P+ Π
P−G
]
(if σ → ∞)(T3.2)
Π
P≡
µ− γ
µY −
W
PNF (T3.3)
P
W= (µk/γ)
(
Y
N
)(1−γ)/γ
(T3.4)
W
P=
ωL1/σ (if σ < ∞)ω (if σ → ∞)
(T3.5)
Notes: ω ≡ γL[αα(1− α)1−α]−1 > 0 and µ ≡ θ/(θ − 1).
Foundations of Modern Macroeconomics - Third Edition Chapter 11 55 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
The flex-price version of the B-K model
Money is neutral: doubling M0 doubles all nominal variables(P,Π,W ) but leaves the real variables(Y,C, L,M0/P,Π/P,W/P ) unaffected
Fiscal policy completely ineffective. There is no income effectin labour supply, so concomitant tax increase does not affectemployment: dY
dG = dLdG = dW
dG = 0 and dCdG = −1 (one-for-one
crowding out of private by public consumption)!
Flex-price B-K model is hyper-classical indeed
Foundations of Modern Macroeconomics - Third Edition Chapter 11 56 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
The menu-cost insight
Small costs of changing one’s actions can have largeallocational and welfare effects
Or: “small deviations from rationality make significantdifferences to equilibria” (Akerlof & Yellen)
In macro-context: following a shock to aggregate demand, isit possible that:
(a) Price stickiness is privately efficient?(b) Price stickiness exists in general equilibrium?(c) Price stickiness has first-order effect on economic welfare?
Foundations of Modern Macroeconomics - Third Edition Chapter 11 57 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Menu-cost insight
In our version of the B-K model we verify the various parts ofthe “menu-cost agenda”:
Part (a) easy: application of the envelope theoremPart (b) tricky: intricate general equilibrium effects(interaction nominal and real rigidity)Part (c) follows once (a)-(b) are covered
Foundations of Modern Macroeconomics - Third Edition Chapter 11 58 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Can it be rational not to change one’s price?
Individual firm cares about its profits only
Optimal price chosen by firm j satisfies:
P ∗j
P=
[
µk
γ·W
P·
(Y
N
)(1−γ)/γ]γ/[γ+θ(1−γ)]
We have combined the optimal pricing rule with the firm’sdemand functionP , Y , and W are all taken as exogenous by the firm (as is N)We can write P ∗
j = P ∗
j (P, Y,W )
The optimized profit function of firm j can be written as:
Π∗j (P, Y,W ) ≡ P ∗
j (·)Yj(P ∗j (·), P, Y
)
−W[
k[Yj
(P ∗j (·), P, Y
)]1/γ+ F
]
Foundations of Modern Macroeconomics - Third Edition Chapter 11 59 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Can it be rational not to change one’s price?
By differentiating Π∗j (·) with respect to aggregate demand, Y ,
we find the envelope result:
dΠ∗
j (·)
dY=
[
[P ∗
j (·)−MC∗
j (·)](∂Yj(Pj , P, Y )
∂Pj
)
Pj=P∗
j
+ Yj(P∗
j (·), P, Y )
]
×
dP ∗
j (·)
dY+
[P ∗
j (·)−MC∗
j (·)] ∂Yj(P
∗
j (·), P, Y )
∂Y
=
[∂Πj(·)
∂Pj
]
Pj=P∗
j
(dP ∗
j (·)
dY
)
+[P ∗
j (·)−MC∗
j (·)] ∂Yj(P
∗
j (·), P, Y )
∂Y
=[P ∗
j (·)−MC∗
j (·)] ∂Yj(P
∗
j (·), P, Y )
∂Y≡
∂Πj(·)
∂Y
Foundations of Modern Macroeconomics - Third Edition Chapter 11 60 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Can it be rational not to change one’s price?
The envelope result:
dΠ∗j (·)
dY=
∂Πj(·)
∂Y
To a first-order of magnitude, the effect on the profit of firm jof a change in aggregate demand is the same whether or notfirm j changes its price optimally following the aggregatedemand shock
Hence, small menu costs will prevent price adjustment by firm
j
Foundations of Modern Macroeconomics - Third Edition Chapter 11 61 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Graphical representation
The menu-cost result is illustrated in Figure 11.3
Initially aggregate demand is Y0 and optimum is at point A
Assume Y rises (to Y1): ceteris paribus (W,P ):
Profit is higher for all Pj and (provided γ < 1) and newoptimum is at point B (north-east from A–output expansionincreases marginal cost)Keeping the old price costs firm j DC in foregone profits. Thisis small because “objective functions are flat at the top”
We have completed part (a) ! Next we work on the GE
repercussions
Foundations of Modern Macroeconomics - Third Edition Chapter 11 62 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
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Non-convex adjustment costsConvex adjustment costs
Figure 11.3: Menu costs
!
!
Pj
A
Aj
Aj(Y0)*
Aj(Y1)*
Pj(Y0)* Pj(Y1)
*
Aj(Pj,P,Y0,WN)0
B
C
D
Aj(Pj,P,Y1,WN)0
Foundations of Modern Macroeconomics - Third Edition Chapter 11 63 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
General equilibrium effects
All firms are in the same position as firm j is in, so they allwant to expand output following an increase in aggregatedemand
Where does the required labour come from?Will there be cost increases because labour is scarce?
Two cases:
σ large (highly elastic labour supply): menu cost equilibriumexistsσ finite/low (moderate labour supply elasticity): generalequilibrium effects destroy menu cost equilibrium (simulationsin Tables 11.4-11.5)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 64 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Table 11.4: Menu costs and the markup
µ = 1.10 µ = 1.25
∆M = 0.05 menu welfare ratio menu welfare ratioσY = 0.1 costs gain costs gain
σ = 0.2 20.44 28.6 1.40 18.10 29.1 1.61σ = 0.5 7.85 28.9 3.68 6.96 29.4 4.22σ = 1 3.95 29.0 7.35 3.51 29.5 8.40σ = 2.5 1.69 29.1 17.18 1.51 29.5 19.49σ = 5 0.94 29.1 30.80 0.86 29.6 34.37σ = 106 0.20 29.1 146.12 0.20 29.6 145.73
Foundations of Modern Macroeconomics - Third Edition Chapter 11 65 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Table 11.4: Menu costs and the markup (continued)
µ = 1.50 µ = 2
σ = 0.2 15.23 29.8 1.96 11.53 30.6 2.65σ = 0.5 5.87 30.0 5.11 4.55 30.8 6.76σ = 1 2.99 30.1 10.06 2.35 30.8 13.12σ = 2.5 1.32 30.1 22.80 1.06 30.8 29.12σ = 5 0.76 30.1 39.56 0.63 30.9 48.68σ = 106 0.21 30.1 144.67 0.21 30.9 144.95
Foundations of Modern Macroeconomics - Third Edition Chapter 11 66 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
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Non-convex adjustment costsConvex adjustment costs
Table 11.5: Menu costs and the elasticity of marginal cost
σY = 0 σY = 0.05
∆M = 0.05 menu welfare ratio menu welfare ratioµ = 1.25 costs gain costs gain
σ = 0.2 17.44 29.2 1.67 17.72 29.2 1.65σ = 0.5 6.61 29.4 4.45 6.76 29.4 4.35σ = 1 3.17 29.5 9.31 3.34 29.5 8.84σ = 2.5 1.19 29.5 24.73 1.36 29.5 21.69σ = 5 0.52 29.6 56.72 0.70 29.6 42.23σ = 106 →0 29.6 → ∞ 0.04 29.6 672.74
Foundations of Modern Macroeconomics - Third Edition Chapter 11 67 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Table 11.5: Menu costs and the elasticity of marginal cost(continued)
σY = 0.1 σY = 0.2
σ = 0.2 18.10 29.1 1.61 18.54 29.1 1.57σ = 0.5 6.96 29.4 4.22 7.34 29.4 4.00σ = 1 3.51 29.5 8.40 3.84 29.5 7.67σ = 2.5 1.51 29.5 19.49 1.83 29.5 16.16σ = 5 0.86 29.6 34.37 1.15 29.5 25.60σ = 106 0.20 29.6 145.73 0.49 29.6 60.60
Foundations of Modern Macroeconomics - Third Edition Chapter 11 68 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
The menu-cost equilibrium (MCE)
Assume σ → ∞ so that the real wage is rigid: if P does notchange (because all firms keep their old prices) then neitherdoes the nominal wage W
Our partial equilibrium story is equivalent to the general
equilibrium effects–see Figure 11.3. Part (b) is confirmed
Properties of the menu cost equilibrium:
Fiscal policy is highly effectiveMonetary policy is highly effective
Both policies have first-order welfare effects. Part (c) is
confirmed
Foundations of Modern Macroeconomics - Third Edition Chapter 11 69 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Fiscal policy in the MCE
in the MCE, the model can be condensed to:
Y = C +G
C =α
1− α
M0
P= α [Y +M0/P −G]
where P is fixed (because all firms keep their price unchanged)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 70 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Fiscal policy in the MCE
Fiscal policy: a lump-sum tax financed increase in G is quiteeffective:
(dY
dG
)MCE
T
= 1
(dC
dG
)MCE
T
=
(d(M0/P )
dG
)MCE
T
= 0
W
P
(dL
dG
)MCE
T
=1
µ
(dY
dG
)MCE
T
=θ − 1
θ> 0
G ↑ causes shift in aggregate demandYj ↑ (for all firms) but Pj (and thus P ) unaffectedLabour supply horizontal, so W unchanged but L ↑Household income rise (both wage income and profitincome)–multiplier effect
Recall the Haavelmo multiplier (from Chapter 1)?
Foundations of Modern Macroeconomics - Third Edition Chapter 11 71 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Monetary policy in the MCE
Helicopter drop of money balances: dM0 > 0 also increasesoutput, consumption, and employment:
P
(dY
dM0
)MCE
= P
(dC
dM0
)MCE
= µW
(dL
dM0
)MCE
=α
1− α> 0
M0 ↑ causes increase in C (wealthier households)Yj ↑ (for all firms) but Pj (and thus P ) unaffectedLabour supply horizontal, so W unchanged, but L ↑Household income rise (both wage income and profitincome)–multiplier effect
Foundations of Modern Macroeconomics - Third Edition Chapter 11 72 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Welfare effects of policy in the MCE
In the menu-cost equilibrium, the hyper-classical modelbecomes hyper-Keynesian (strong effects of policy)
But: what are the welfare effects of fiscal and monetarypolicy?
Foundations of Modern Macroeconomics - Third Edition Chapter 11 73 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Welfare effects of policy in the MCE
Indirect utility function (IUF):
V = αα(1− α)1−α
[
Y +M0
P−G
]
− γLL
= αα(1− α)1−α
[M0 +Π
P−G
]
+
[
αα(1− α)1−α
(W
P
)
− γL
]
L
= αα(1− α)1−α
[M0 +Π
P−G
]
(a)
Used Π ≡ PY −WL in the first stepUsed labour supply, γL = αα(1− α)1−α
(WP
), in second step:
labour supply set optimally so variation in L causes nofirst-order welfare effect
From (a) we conclude that both policies cause first-orderwelfare effect
Foundations of Modern Macroeconomics - Third Edition Chapter 11 74 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Welfare effects of policy in the MCE
Fiscal policy:
(dV
dG
)MCE
T
= αα(1− α)1−α
[(dY
dG
)MCE
T
− 1
]
− γL
(dL
dG
)MCE
T
= −γLµ
(P
W
)
= −αα(1− α)1−α
µ< 0
dYdG = 1 does not come for free as the household must supplymore hours of labourNet effect on welfare is negative
Foundations of Modern Macroeconomics - Third Edition Chapter 11 75 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Welfare effects of policy in the MCE
Monetary policy:(
dV
dM0
)MCE
= αα(1− α)1−α
[
1
P+
(d(Π/P )
dM0
)MCE]
=αα(1− α)1−α
P
[
1 + P
(dY
dM0
)MCE
−W
(dL
dM0
)MCE]
=αα(1− α)1−α
P︸ ︷︷ ︸
(a)
[
1 +1
θ
α
1− α
]
︸ ︷︷ ︸
(b)
> 0
(a): marginal utility of nominal income (positive)(b), first term: liquidity effect (also in competitive model)(b), second term: profit effect (specific to B-K model)Total effect on welfare is positive, more so the larger is thedegree of monopoly ( 1θ )
The liquidity effect holds because the competitive monetary
equilibrium is sub-optimal: Friedman’s satiation condition violated
Foundations of Modern Macroeconomics - Third Edition Chapter 11 76 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
The general MCE equilibrium
Now we assume that labour supply features a finite elasticity:0 < σ ≪ ∞
Analytical results no longer possible: GE effects toocomplicated
Calibrate the model and simulate robustness of menu-costresult to variations in:
The value of σThe value of µ (the gross monopoly markup)The value of σY ≡ 1−γ
γ (the elasticity of the marginal cost
function)
The calibration exercise allows the evaluation of big shocks(inframarginal)
Assume that the menu costs take the form of labour input Z(e.g. shop assistants changing price tags)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 77 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
The general MCE equilibrium
Following a monetary shock there are two scenario’s:
(FA) full adjustment: all firms change their price and incur themenu costs
ΠFA =µ− γ
µPY −WN (F + Z)
(NA) non-adjustment: all firms keep their price unchanged
ΠNA = P0Y −W[
kY 1/γN1−1/γ +NF]
Foundations of Modern Macroeconomics - Third Edition Chapter 11 78 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
The general MCE equilibrium
In the simulations we find minimum value of Z (labeledZMIN ) for which non-adjustment is an equilibrium (for whichΠNA > ΠFA). See Tables 11.4-11.5
The entry “menu costs” is defined as follows:
menu costs = 100×
[
N0 (W )NA
ZMIN
P0Y NA
]
The entry “welfare gain” is defined as follows:
welfare gain = 100×
[V NA − V0
UCY NA
]
The entry “ratio” is defined as welfare costmenu cost
Foundations of Modern Macroeconomics - Third Edition Chapter 11 79 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
The general MCE equilibrium
Example from Table 11.4: µ = 1.1, σY = 0.1, and σ = 106.Menu costs amounting to no more than 0.20% of revenue(tiny) will make non-adjustment of prices an equilibrium inthe sense that ΠNA > ΠFA! The welfare effect is 29.1% ofoutput (huge). Small menu costs have large welfare effects
Other key features of the simulation results:
Welfare measure relatively constantThe markup does not affect menu costs and ratio very muchThe labour supply elasticity exerts a very strong effect onmenu costs and the ratio. Intuition: if σ is low, then outputexpansion drives up wages (production costs) which makesnon-adjustment less likely to be optimal
Table 11.5 has basically very similar results: the key role isplayed by the labour supply elasticity
Foundations of Modern Macroeconomics - Third Edition Chapter 11 80 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Evaluation of the menu-cost idea
Runs into same trouble as the RBC literature does: we simplydo not observe a high σ
Ball & Romer: both nominal rigidity (menu cost) and somekind of real rigidity (e.g. high σ, customer market, orefficiency wage labour market) are needed to get themenu-cost equilibrium
Rotemberg mentions some further problems:
MC equilibrium may not be uniqueMay equally well apply to quantities instead of prices (makesprice adjustment more likely)MC insight does not generalize easily to dynamic setting (ournext theories do not have that problem)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 81 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Quadratic price adjustment costs
Convex adjustment costs: quadratic in price change
Derive approximate pricing rule in two steps:
Determine path of equilibrium prices P ∗
j,τ∞
τ=0 which the firmwould set in the absence of price-adjustment costs (PACs).This is the desired “target” the firm will aim forNext determine the quadratic approximation of the profitfunction around this target price path and incorporate PACs
Foundations of Modern Macroeconomics - Third Edition Chapter 11 83 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Quadratic price adjustment costs
The objective function of the firm is then:
Ω0 =
∞∑
τ=0
(1
1 + ρ
)τ
(pj,τ − p∗j,τ
)2
︸ ︷︷ ︸
(a)
+ c (pj,τ − pj,τ−1)2
︸ ︷︷ ︸
(b)
Stay as close as possible to target path: Ω0 should beminimizedpj,τ ≡ logPj,τ (actual price); p∗j,τ ≡ logP ∗
j,τ (target price)ρ is the firm’s discount factor(a): intratemporal cost of setting the “wrong” price(b): intertemporal costs associated with changing the price(annoyed customers, etcetera)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 84 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Quadratic price adjustment costs
The firm minimizes Ω0 by choosing the appropriate sequenceof prices, pj,τ
∞τ=0. The FONC is:
∂Ω0
∂pj,τ=
(1
1 + ρ
)τ[2(pj,τ − p∗j,τ
)+ 2c (pj,τ − pj,τ−1)
]
−
(1
1 + ρ
)τ+1
[2c (pj,τ+1 − pj,τ )] = 0
or:
pj,τ+1−
[
1 + (1 + ρ)1 + c
c
]
pj,τ +(1+ ρ)pj,τ−1 = −1 + ρ
cp∗j,τ
(a)Equation (a) is a second-order difference equation is pj,τ Weneed two boundary conditions:
Initial condition: pj,−1 is pre-determined (set in the past)Terminal condition
Foundations of Modern Macroeconomics - Third Edition Chapter 11 85 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Quadratic price adjustment costs
The pricing rule in the planning period (pj,0) is then:
pj,0 = λ1pj,−1 + (1− λ1)
[
λ2 − 1
λ2
∞∑
τ=0
(1
λ2
)τ
p∗j,τ
]
(b)
0 < λ1 < 1 is the stable characteristic root of (a)λ2 > 1 is the unstable characteristic root of (a)Actual price weighted average of the past price and a long-runtarget priceNote that (b) contains both backward-looking andforward-looking elements. Anticipated changes in p∗j,τ willimmediately have an effect on the current price
Foundations of Modern Macroeconomics - Third Edition Chapter 11 86 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Slaggered price setting
Guillermo Calvo (and co-workers) have devised an alternativeapproach to price stickiness (red light-green light model)
Price contracts are staggered (old idea of Phelps and Taylor)
No separate price-adjustment costs
Duration of price contract is stochastic via a Poisson process:each period “nature” draws a signal to each firm:
“green light” with probability π: go ahead and adjust yourcontract price (optimally)“red light” with probability 1− π: continue to charge yourpresent contract price
Foundations of Modern Macroeconomics - Third Edition Chapter 11 87 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Slaggered price setting
Objective function of a firm which has just received a greenlight:
Ω0 =(pj,0 − p∗j,0
)2+
1
1 + ρ
[
π(pj,1 − p∗j,1
)2+ (1− π)
(pj,0 − p∗j,1
)2]
+
(1
1 + ρ
)2[π2
(pj,2 − p∗j,2
)2+ π(1− π)
(pj,1 − p∗j,2
)2
+ (1− π)2(pj,0 − p∗j,2
)2]+ higher-order terms
Period τ = 0: you can set your price at pj,0 (takeintratemporal costs into account)Period τ = 1: you may get a green or a red light. In lattercase, you keep old price, pj,0. In the former case, you canre-optimize and determine pj,1Period τ = 2: three possibilities . . .
Foundations of Modern Macroeconomics - Third Edition Chapter 11 88 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Slaggered price setting
Collecting terms involving pj,0 we get:
Ω0 =(pj,0 − p∗j,0
)2+
1− π
1 + ρ
(pj,0 − p∗j,1
)2+
(1− π
1 + ρ
)2(pj,0 − p∗j,2
)2+ . . .
=∞∑
τ=0
(1− π
1 + ρ
)τ(pj,0 − p∗j,τ
)2+ uninteresting terms (a)
Pricing friction shows up as heavier discounting: if π ≈ 1 youhave almost perfect price flexibility. If π ≈ 0 you attach higherweight to future deviation costs
The firm chooses pj,0 in order to minimize Ω0. The FONC is:
pj,0
∞∑
τ=0
(1− π
1 + ρ
)τ
=∞∑
τ=0
(1− π
1 + ρ
)τ
p∗j,τ
Foundations of Modern Macroeconomics - Third Edition Chapter 11 89 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Slaggered price setting
We get:
pn0 =π + ρ
1 + ρ
∞∑
τ=0
(1− π
1 + ρ
)τ
p∗τ (new price)
Firms facing a red light maintain their old prices:
pn−s =π + ρ
1 + ρ
∞∑
τ=0
(1− π
1 + ρ
)τ
p∗τ−s (price set s period ago)
Given the Poisson process and the assumption of a largenumber of firms we know that π(1− π)s is the fraction offirms which last set its price s periods ago
Foundations of Modern Macroeconomics - Third Edition Chapter 11 90 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Staggered price setting
We can aggregate all prices to derive an expression for theaggregate price level:
p0 = πpn0 + π(1− π)pn−1 + π(1− π)2pn−2 + π(1− π)3pn−3 + . . .
= π∞∑
s=0
(1− π)spn−s
= πpn0 + (1− π)p−1
Substitute the expression for pn0 :
p0 = (1− π)p−1 + π
[
π + ρ
1 + ρ
∞∑
τ=0
(1− π
1 + ρ
)τ
p∗τ
]
Foundations of Modern Macroeconomics - Third Edition Chapter 11 91 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Staggered price setting
The aggregate price level:
p0 = (1− π)p−1 + π
[
π + ρ
1 + ρ
∞∑
τ=0
(1− π
1 + ρ
)τ
p∗τ
]
Actual price is weighed average of new price and past priceRotemberg and Calvo approaches “observationally equivalent”(yield same macro pricing equation)Rotemberg estimates that in the US 8% of all prices areadjusted each quarter (mean time between price adjustments isthree years)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 92 / 93
‘Keynesian’ MultipliersMonopolistic competition and money
Imperfectly flexible prices
Non-convex adjustment costsConvex adjustment costs
Punchlines
General equilibrium monopolistic competition (MC) modelprovides micro-foundations for multiplier
Intimate link between multplier and welfare effects(pre-existing distortion)
The existence of MC does not render money neutral! Weneed price-adjustment costs
The menu cost insight: small deviations from rationality canhave large macroeconomic and welfare effects (need bothnominal and real rigidity)
Practical models: convex adjustment costs (macroeconomicprice level becomes backward-looking state variable)
Foundations of Modern Macroeconomics - Third Edition Chapter 11 93 / 93